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Coated inclusions of finite conductivity neutral to multiple fields in two-dimensional conductivity or anti-plane elasticity

Published online by Cambridge University Press:  10 March 2014

HYEONBAE KANG  and
HYUNDAE LEE
HYEONBAE KANG
Affiliation:
Department of Mathematics, Inha University, Incheon 402-751, South Korea email: hbkang@inha.ac.kr
HYUNDAE LEE
Affiliation:
Department of Mathematics, Inha University, Incheon 402-751, South Korea email: hbkang@inha.ac.kr
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Abstract

We consider the problem of neutral inclusions for two-dimensional conductivity and anti-plane elasticity. The neutral inclusion, when inserted in a matrix having a uniform field, does not disturb the field outside the inclusion. The inclusion consists of a core and a shell. We show that if the inclusion is neutral to two linearly independent fields, then the core and the shell are confocal ellipses.

Keywords

neutral inclusionfinite conductivitycoated inclusion
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 3 , June 2014 , pp. 329 - 338
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Ammari, H., Garnier, J., Jugnon, V., Kang, H., Lee, H. & Lim, M. (2012) Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions. Contemp. Math. 577, 1–24.Google Scholar
[2]Ammari, H. & Kang, H. (2007) Polarization and moment tensors with applications to inverse problems and effective medium theory. In: Applied Mathematical Sciences, Vol. 162, Springer-Verlag, New York, NY.Google Scholar
[3]Ammari, H., Kang, H., Lee, H. & Lim, M. (2013) Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem. Comm. Math. Phys. 317, 253–266.Google Scholar
[4]Ammari, H., Kang, H., Lee, H. & Lim, M. (2013) Enhancement of near-cloaking. Part II: The Helmholtz equation. Comm. Math. Phys. 317, 485–502.Google Scholar
[5]Ammari, H., Kang, H., Lee, H., Lim, M. and Yu, S. (2013) Enhancement of near cloaking for the full Maxwell equations. SIAM J. Appl. Math. 73, 2055–2076.Google Scholar
[6]Dive, P. (1931) Attraction des ellipsoides homogènes et réciproques d'un théorème de Newton. Bull. Soc. Math. France 59, 128–140.Google Scholar
[7]Greenleaf, A., Lassas, M. & Uhlmann, G. (2003) On non-uniqueness for Calderon's inverse problem. Math. Res. Lett. 10, 685–693.Google Scholar
[8]Hashin, Z. (1962) The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150.Google Scholar
[9]Hashin, Z. & Shtrikman, S. (1962) A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phy. 33, 3125–3131.Google Scholar
[10]Jarczyk, P. & Mityushev, V. (2012) Neutral coated inclusions of finite conductivity. Proc. R. Soc. A 468, 954–970.Google Scholar
[11]Kang, H. (2009) Conjectures of Polya-Szego and Eshelby, and the Newtonian potential problem: A review. Mech. Mater. 41, 405–410.Google Scholar
[12]Kang, H. & Milton, G. W. (2008) Solutions to the Pólya-Szegö conjecture and the weak Eshelby conjecture. Arch. Rational Mech. Anal. 188, 93–116.Google Scholar
[13]Kerker, M. (1975) Invisible bodies. J. Opt. Soc. Am. 65, 376–379.CrossRefGoogle Scholar
[14]Kohn, R. V., Onofrei, D., Vogelius, M. S. & Weinstein, M. I. (2010) Cloaking via change of variables for the Helmholtz equation. Comm. Pure Appl. Math. 63, 973–1016.CrossRefGoogle Scholar
[15]Kohn, R. V., Shen, H., Vogelius, M. S. & Weinstein, M. I. (2008) Cloaking Via Change Of Variables In Electric Impedance Tomography. Inverse Probl. 24, article 015016.Google Scholar
[16]Milton, G. W. (2002) The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK.Google Scholar
[17]Milton, G. W. & Serkov, S. K. (2001) Neutral coated inclusions in conductivity and anti-plane elasticity. Proc. R. Soc. Lond. A 457, 1973–1997.Google Scholar
[18]Nikliborc, W. (1932) Eine Bemerkung über die Volumpotentiale. Math. Zeit. 35, 625–631.Google Scholar
[19]Pendry, J. B., Schurig, D. & Smith, D. R. (2006) Controlling electromagnetic fields. Science 312, 1780–1782.Google Scholar